Multifractal models for asset prices

In this paper, we make a short overview of multifractal models of asset returns. All the proposed models rely upon the notion of random multiplicative cascades. We focus in more details on the simplest of such models namely the log-normal Multifractal Random Walk. This model can be seen as a stochastic volatility model where the (log-) volatility has a peculiar long-range correlated memory. We briefly address calibration issues of such models and their applications to volatility and VaR forecasting. Since Mandelbrot first work on the fluctuations of cotton price in early sixties, it is well known that market price variations are poorly described by the standard geometric Brownian motion [15]: Extreme events are more probable than in a Gaussian world and volatility fluctuations are well kwown to be of intermittent and correlated nature. As we shall discuss along this article, multifractal analyis has provided new concepts and tools to analyze market fluctuations and inspired a particularly elegant family of models that ∗This paper was written in the framework of the Chair Financial Risks of the Risk Foundation sponsored by Société Générale.